A village P is 12km from village Q. If it takes 3 hours 20 minutes to travel from Q to P and back to Q by a boat. If the boat travels at a speed of 6km/h from P to Q and (6+x)km/h back to P, find the value of x.
A village P is 12km from village Q. If it takes 3 hours 20 minutes to travel from Q to P and back to Q by a boat. If the boat travels at a speed of 6km/h from P to Q and (6+x)km/h back to P, find the value of x.
since time = distance/speed,
12/6 + 12/(6+x) = 10/3
12/(6+x) = 4/3
4(6+x) = 12*3
x = 3
So it takes 1 hour and 50 minutes 1 way and it takes 2 hours to do the first way, so in order to go back the other way they went faster.
Hope this helps and wasn't too confusing :)
Also I'm not a math genius so if this is wrong I'm sorry
I forgot to say this in my last one, but you still need to figure out how much faster they were going
Well, you see, the boat seems to be quite indecisive about its speed. It's like it can't make up its mind whether it wants to go fast or slow. I mean, going at 6km/h from P to Q and then increasing the speed by x km/h on the way back to P? That's just playing mind games with the poor boat.
But fear not, for I am here to help. Let's break down the problem. The boat takes 3 hours 20 minutes to travel from Q to P and back to Q. Now, if we assume that the boat travels at a speed of 6km/h from P to Q, then the distance from P to Q would be covered in 12/6 = 2 hours.
So, the total time it takes for the boat to go from P to Q and back to P would be 2 + (2 + (x/6)) = 3 hours 20 minutes. And since 3 hours 20 minutes is equivalent to 3 + (20/60) = 3.33 hours, we can write the equation as:
2 + (2 + (x/6)) = 3.33
Now, let's solve for x. Simplifying the equation, we get:
4 + (x/6) = 3.33
Subtracting 4 from both sides gives us:
x/6 = -0.67
Finally, multiplying both sides by 6, we find:
x = -4.02
So, according to my calculations, the value of x is approximately -4.02. However, this negative value doesn't make much sense in this context. It seems our boat is not only indecisive but also has a hidden talent for going backward. Therefore, I would advise against taking this boat and instead, consider finding a more reliable means of transportation.
To find the value of x, we need to determine the time it takes to travel from P to Q and back to P, using the given information.
Let's break down the journey into two parts:
1. Travel from P to Q:
The distance from P to Q is 12 km, and the boat travels at a speed of 6 km/h. To find the time taken, we can use the formula: Time = Distance / Speed.
So, the time taken to travel from P to Q is:
Time_pq = Distance_pq / Speed_pq = 12 km / 6 km/h = 2 hours.
2. Travel from Q back to P:
We know that the time taken for the whole trip (from Q to P and back to Q) is 3 hours 20 minutes, which is equivalent to 3 + 20/60 hours = 3.33 hours.
Therefore, the time taken for the journey from Q to P and back to Q is:
Time_qpq = Time_pq + Time_return.
Since the return journey is made at a speed of (6 + x) km/h, we can calculate the time taken for the return journey using the same formula:
Time_return = Distance_pq / Speed_return = 12 km / (6 + x) km/h = 2 hours.
Now, let's substitute the values into the equation for the total travel time:
Time_qpq = Time_pq + Time_return = 2 hours + 2 hours = 4 hours.
However, we are given that the total time taken is 3.33 hours. This discrepancy means that our initial assumption for the return speed (6 + x) km/h is incorrect.
To find the correct return speed, we can use the formula for average speed:
Average Speed = Total Distance / Total Time.
The total distance for the round trip is 2 * Distance_pq = 2 * 12 km = 24 km.
Substituting the values into the average speed formula:
Average Speed = Total Distance / Total Time = 24 km / 3.33 hours.
To find the correct return speed, we need to consider that the boat is traveling half the distance at 6 km/h and the other half at (6 + x) km/h.
Therefore, we can set up an equation:
Average Speed = (1/2) * Speed_pq + (1/2) * Speed_return.
Substituting the known values into the equation:
24 km / 3.33 hours = (1/2) * 6 km/h + (1/2) * (6 + x) km/h.
Now we can solve for x:
24 km / 3.33 hours = 3 km/h + (6 + x)/2 km/h.
Multiplying both sides by 3.33 hours to remove the denominator:
24 km = 3 * 3.33 + (6 + x)/2 * 3.33 km.
Simplifying further:
24 km = 9.99 km + (6 + x) * 1.665 km.
24 km - 9.99 km = 9.99 km + 1.665x km.
14.01 km = 9.99 km + 1.665x km.
Subtracting 9.99 km from both sides:
14.01 km - 9.99 km = 1.665x km.
4.02 km = 1.665x km.
To find x, divide both sides of the equation by 1.665 km:
x = (4.02 km) / (1.665 km).
x ≈ 2.415.
Therefore, the value of x is approximately 2.415 km/h.